Existence and Stability of Standing Pulses in Neural Networks: I. Existence

Abstract

We consider the existence of standing pulse solutions of a neural network integrodifferential equation. These pulses are bistable with the zero state and may be an analogue for short term memory in the brain. The network consists of a single-layer of neurons synaptically connected by lateral inhibition. Our work extends the classic Amari result by considering a non-saturating gain function. We consider a specific connectivity function where the existence conditions for singlepulses can be reduced to the solution of an algebraic system. In addition to the two localized pulse solutions found by Amari, we find that three or more pulses can coexist. We also show the existence of nonconvex "dimpled" pulses and double pulses. We map out the pulse shapes and maximum firing rates for different connection weights and gain functions.

Exploiting the Hamiltonian structure of a neural field model

“…The model (1)- (3), and other similar models [6,10,24,33], have been used to investigate working memory, as stable single-bump solutions are thought to be the analogue of short-term memory. We have demonstrated here how a specific model can lose stable single-bump solutions as parameters are varied.…”

“…Solutions of (9) which are homoclinic to the origin correspond to spatially-localised steady states of (1). This technique of using Fourier transforms to convert integral equations to differential equations has been used several times before [6,17,[23][24][25][26]. Equation (9) has a number of important properties which we now discuss.…”

“…More analytical progress can be made if the firing rate function f is replaced by a piecewise linear function [6] or a Heaviside step function [8,24]. Here we consider the case when f (u) = 2Θ(u − θ), which is the result of taking r → 0 in (2).…”

“…Spatially localised solutions are of importance in many different areas, such as the study of localised buckling of long struts [1,2], nonlinear optics [3], vibrating granular media [4], convection problems [5] and neuroscience [6,7]. In neural field models, stationary spatially localised regions of high activity ("bumps") have been studied in the context of working memory, as single-bump steady state solutions are believed to be the analogue of shortterm memory [6,[8][9][10]. Although these systems are quite diverse, they often display similar behaviour.…”

Waves, bumps, and patterns in neural field theories

“…Another common choice of w(x) in the study of neural field models is that of a Mexican hat function, such as w(x) = (1 − |x|)e −|x| /4 (perhaps more properly called a wizard hat function, because of its cusp at the origin [35]). In this case…”